This thesis studies the mathematical behavior of random polynomials in terms of the expected number of real zeros and the exceedance measure. To this end, the important formulae for studying this behavior are reviewed, generalized and applied to random polynomials of three different types: the polynomials with symmetric coefficients, the amplified polynomials and the de-amplified polynomials. By defining the coefficients a/ s from the classic form of polynomial P(x) = L 7=oa jxi as independently normally distributed random variables with two-fold symmetry, aj = a-n-j-1 a polynomial with symmetric coefficients is studied. Then by introducing binomial factors, (n,j) 1/2, into coefficients, the types of amplified and de-amplified polynomials are defined. Besides the expected number of real zeros, the exceedance measure is considered for better understanding of the behavior of amplified and de-amplified polynomials. Later in this thesis a valued progress is made towards a conjecture that constants been missed from the sum of binomial series have insignificant impact on the expected number of real zeros. The early studies in this thesis implies that the binomial factors facilitate the evaluation of the behavior, therefore a more difficult class of random polynomial without binomial factors is studied as final part of work of the thesis and supported by numerical analysis
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:591075 |
| Date | January 2012 |
| Creators | Gao, Jianliang |
| Publisher | Ulster University |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
Page generated in 0.0016 seconds